English Usage: Anyone Versus Everyone

in Ex-Facebook

Once, on Facebook I wrote:

Here’s a tough (English) linguistics question I’m chasing down: What is the difference between “anyone” and “everyone?” Anyone means that “For all X, given any arbitrary X, some quality is the case.” Everyone means that for all X they exhibit some quality.

I’m not sure that there’s a satisfactory answer in this link, but it’s something I’m interested in understanding.